| L(s) = 1 | + 2·7-s + 11-s + 4·17-s + 4·19-s + 6·23-s − 2·29-s + 8·31-s − 4·37-s + 6·41-s − 6·43-s + 2·47-s − 3·49-s + 12·53-s − 4·59-s + 14·61-s + 10·67-s − 8·71-s + 4·73-s + 2·77-s − 8·79-s − 2·83-s + 10·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.970·17-s + 0.917·19-s + 1.25·23-s − 0.371·29-s + 1.43·31-s − 0.657·37-s + 0.937·41-s − 0.914·43-s + 0.291·47-s − 3/7·49-s + 1.64·53-s − 0.520·59-s + 1.79·61-s + 1.22·67-s − 0.949·71-s + 0.468·73-s + 0.227·77-s − 0.900·79-s − 0.219·83-s + 1.05·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.150822060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.150822060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.71401715392406, −14.99024426074066, −14.62883639312969, −14.09903853333525, −13.55128919186779, −12.98911407923050, −12.28927458600986, −11.73582486681754, −11.41268340036347, −10.70982105596731, −10.08092396434632, −9.595284920028108, −8.893669246657166, −8.322690027246134, −7.781877509559944, −7.131749796001232, −6.633495792326322, −5.679377531026011, −5.260885896932720, −4.639748489688625, −3.834741111514256, −3.153744158610845, −2.407521793081302, −1.386709604045879, −0.8272719222973923,
0.8272719222973923, 1.386709604045879, 2.407521793081302, 3.153744158610845, 3.834741111514256, 4.639748489688625, 5.260885896932720, 5.679377531026011, 6.633495792326322, 7.131749796001232, 7.781877509559944, 8.322690027246134, 8.893669246657166, 9.595284920028108, 10.08092396434632, 10.70982105596731, 11.41268340036347, 11.73582486681754, 12.28927458600986, 12.98911407923050, 13.55128919186779, 14.09903853333525, 14.62883639312969, 14.99024426074066, 15.71401715392406
